Atomistic and Continuum Models of Solids

E0708026研究人员继续致力于从第一原理开始发展结晶固体的系统数学理论，重点是了解原子学/电子结构模型和连续介质模型之间的联系。他和他的合作者研究了电子结构模型的连续极限，例如Kohn-Sham密度泛函理论中的模型。他们还研究了结合Kohn-Sham密度泛函理论和连续介质模型的数值算法。特别是，他们研究这些算法的精度，并找到实现统一精度的策略。这些问题对于理解电子结构模型并使其成为材料科学的强大而实用的工具至关重要。该项目有助于为固体的连续力学提供微观基础。从更广泛的角度来看，该项目是为建立多尺度、多物理模型的基础而需要做的工作的一个例子。现在广泛的科学学科普遍认为，下一个重大进展将来自多尺度、多物理建模，它将量子力学的模型与经典或连续力学的模型相结合。然而，由于缺乏坚实的基础，目前在多尺度、多物理模拟方面的许多活动本质上是临时的。这种做法违背了多尺度、多物理建模的初衷。这个项目的目的是为晶体固体的情况建立必要的基础。此外，对纳米技术和基于第一原理的材料设计的兴趣要求更好地理解固体的机械性能和它们的电子结构之间的联系。在连续介质理论中经常使用的传统概念将不得不修改，以便考虑到电子结构水平上的影响。建立这种联系是拟议项目的一个主要重点。

E0708026 The investigator continues work on developing a systematicmathematical theory of crystalline solids starting from firstprinciples, with emphasis on understanding the connection betweenatomistic/electronic structure models and continuum models. Heand his collaborators study the continuum limit of electronicstructure models such as models in Kohn-Sham density functionaltheory. They also study numerical algorithms that combineKohn-Sham density functional theory and continuum models. Inparticular, they study the accuracy of these algorithms and findstrategies for achieving uniform accuracy. These issues arecritical for understanding the electronic structure models andmaking them powerful and practical tools for material science. The project helps to provide the microscopic foundation of thecontinuum mechanics of solids. From a broader perspective, the project serves as an exampleof the kind of work that needs to be done in order to establishthe foundation for multi-scale, multi-physics modeling. It isnow generally recognized across a wide spectrum of scientificdisciplines that the next major advances will come frommulti-scale, multi-physics modeling that integrates models fromquantum mechanics with models from classical or continuummechanics. However, due to the lack of a solid foundation, muchof the current activities in multi-scale, multi-physics modelingare ad hoc in nature. Such a practice defeats the originalpurpose of multi-scale, multi-physics modeling. This project isaimed at establishing the necessary foundation for the case ofcrystalline solids. In addition, interest in nanotechnology andfirst-principle-based material design demands a much betterunderstanding of the connection between the mechanical propertiesof solids and their electronic structure. Traditional conceptsthat are often used in continuum theory will have to be modifiedin order to take into account the effects at the electronicstructure level. Establishing such a connection is a major focusof the proposed project.